第1页 / 共316页
第2页 / 共316页
第3页 / 共316页
第4页 / 共316页
第5页 / 共316页
第6页 / 共316页
第7页 / 共316页
第8页 / 共316页
Bayesian Data Analysis in Ecology Using Linear Models with R, BU....pdf
Bayesian Data Analysis in Ecology Using Linear Models with R, BUGS, and Stan
Copyright
Digital Assets
Acknowledgments
1. Why do we Need Statistical Models and What is this Book About?
1.1 Why We Need Statistical Models
1.2 What This Book is About
Further Reading
2. Prerequisites and Vocabulary
2.1 Software
2.1.1 What Is R?
2.1.2 Working with R
2.2 Important Statistical Terms and How to Handle Them in R
2.2.1 Data Sets, Variables, and Observations
2.2.2 Distributions and Summary Statistics
2.2.3 More on R Objects
2.2.4 R Functions for Graphics
2.2.5 Writing Our Own R Functions
Further Reading
3. The Bayesian and the Frequentist Ways of Analyzing Data
3.1 Short Historical Overview
3.2 The Bayesian Way
3.2.1 Estimating the Mean of a Normal Distribution with a Known Variance
3.2.2 Estimating Mean and Variance of a Normal Distribution Using Simulation
3.3 The Frequentist Way
3.4 Comparison of the Bayesian and the Frequentist Ways
Further Reading
4. Normal Linear Models
4.1 Linear Regression
4.1.1 Background
4.1.2 Fitting a Linear Regression in R
4.1.3 Drawing Conclusions
4.1.4 Frequentist Results
4.2 Regression Variants: ANOVA, ANCOVA, and Multiple Regression
4.2.1 One-Way ANOVA
4.2.2 Frequentist Results from a One-Way ANOVA
4.2.3 Two-Way ANOVA
4.2.4 Frequentist Results from a Two-Way ANOVA
4.2.5 Multiple Comparisons and Post Hoc Tests
4.2.6 Analysis of Covariance
4.2.7 Multiple Regression and Collinearity
4.2.8 Ordered Factors and Contrasts
4.2.9 Quadratic and Higher Polynomial Terms
Further Reading
5. Likelihood
5.1 Theory
5.2 The Maximum Likelihood Method
5.3 The Log Pointwise Predictive Density
Further Reading
6. Assessing Model Assumptions: Residual Analysis
6.1 Model Assumptions
6.2 Independent and Identically Distributed
6.3 The QQ Plot
6.4 Temporal Autocorrelation
6.5 Spatial Autocorrelation
6.6 Heteroscedasticity
Further Reading
7. Linear Mixed Effects Models
7.1 Background
7.1.1 Why Mixed Effects Models?
7.1.2 Random Factors and Partial Pooling
7.2 Fitting a Linear Mixed Model in R
7.3 Restricted Maximum Likelihood Estimation
7.4 Assessing Model Assumptions
7.5 Drawing Conclusions
7.6 Frequentist Results
7.7 Random Intercept and Random Slope
7.8 Nested and Crossed Random Effects
7.9 Model Selection in Mixed Models
Further Reading
8. Generalized Linear Models
8.1 Background
8.2 Binomial Model
8.2.1 Background
8.2.2 Fitting a Binomial Model in R
8.2.3 Assessing Model Assumptions: Overdispersion and Zero-Inflation
Option 1
Option 2
8.2.4 Drawing Conclusions
8.2.5 Frequentist Results
8.3 Fitting a Binary Logistic Regression in R
8.3.1 Some Final Remarks
8.4 Poisson Model
8.4.1 Background
8.4.2 Fitting a Poisson-Model in R
8.4.3 Assessing Model Assumptions
8.4.4 Drawing Conclusions
8.4.5 Modeling Rates and Densities: Poisson Model with an Offset
8.4.6 Frequentist Results
Further Reading
9. Generalized Linear Mixed Models
9.1 Binomial Mixed Model
9.1.1 Background
9.1.2 Fitting a Binomial Mixed Model in R
9.1.3 Assessing Model Assumptions
9.1.4 Drawing Conclusions
9.2 Poisson Mixed Model
9.2.1 Background
9.2.2 Fitting a Poisson Mixed Model in R
9.2.3 Assessing Model Assumptions
9.2.4 Drawing Conclusions
9.2.5 Modeling Bird Densities by a Poisson Mixed Model Including an Offset
Further Reading
10. Posterior Predictive Model Checking and Proportion of Explained Variance
10.1 Posterior Predictive Model Checking
10.2 Measures of Explained Variance
Further Reading
11. Model Selection and Multimodel Inference
11.1 When and Why We Select Models and Why This is Difficult
11.2 Methods for Model Selection and Model Comparisons
11.2.1 Cross-Validation
11.2.2 Information Criteria: Akaike Information Criterion and Widely Applicable Information Criterion
11.2.3 Other Information Criteria
11.2.4 Bayes Factors and Posterior Model Probabilities
11.2.5 Model-Based Methods to Obtain Posterior Model Probabilities and Inclusion Probabilities
11.2.6 “Least Absolute Shrinkage and Selection Operator” 䰀䄀匀匀伀 and Ridge Regression
11.3 Multimodel Inference
11.4 Which Method to Choose and Which Strategy to Follow
Further Reading
12. Markov Chain Monte Carlo Simulation
12.1 Background
12.2 MCMC Using BUGS
12.2.1 Using BUGS from OpenBUGS
12.2.2 Using BUGS from R
12.3 MCMC Using Stan
12.4 Sim, BUGS, and Stan
Further Reading
13. Modeling Spatial Data Using GLMM
13.1 Background
13.2 Modeling Assumptions
13.3 Explicit Modeling of Spatial Autocorrelation
13.3.1 Starting the Model Fitting
13.3.2 Variogram Modeling
13.3.3 Bayesian Modeling
13.3.4 OpenBUGS Example
Further Reading
14. Advanced Ecological Models
14.1 Hierarchical Multinomial Model to Analyze Habitat Selection Using BUGS
14.2 Zero-Inflated Poisson Mixed Model for Analyzing Breeding Success Using Stan
14.3 Occupancy Model to Measure Species Distribution Using Stan
14.4 Territory Occupancy Model to Estimate Survival Using BUGS
14.5 Analyzing Survival Based on Mark-Recapture Data Using Stan
Further Reading
15. Prior Influence and Parameter Estimability
15.1 How to Specify Prior Distributions
15.2 Prior Sensitivity Analysis
15.3 Parameter Estimability
Further Reading
16. Checklist
16.1 Data Analysis Step by Step
Step 1: Plausibility of Data
Step 2: Relationships
Step 3: Error Distribution
Step 4: Preparation of Explanatory Variables
Step 5: Data Structure
Step 6: Fit the Model
Step 7: Check Model Assumptions, Fit, and Sensitivity
Step 8: Model Uncertainty
Step 9: Draw Conclusions
Further Reading
17. What Should I Report in a Paper
17.1 How to Present the Results
17.2 How to Write Up the Statistical Methods
Further Reading
References
Index
Bayesian Data Analysis
in Ecology Using Linear
Models with R, BUGS,
and Stan
Fra¨ nzi Korner-Nievergelt
Tobias Roth
Stefanie von Felten
Je´ roˆ me Gue´ lat
Bettina Almasi
Pius Korner-Nievergelt
AMSTERDAM l BOSTON l HEIDELBERG l LONDON
NEW YORK l OXFORD l PARIS l SAN DIEGO
SAN FRANCISCO l SINGAPORE l SYDNEY l TOKYO
Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier
32 Jamestown Road, London NW1 7BY, UK
525 B Street, Suite 1800, San Diego, CA 92101-4495, USA
225 Wyman Street, Waltham, MA 02451, USA
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK
Copyright Ó 2015 Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by
any means, electronic or mechanical, including photocopying, recording, or any
information storage and retrieval system, without permission in writing from the
publisher. Details on how to seek permission, further information about the Publisher’s
permissions policies and our arrangement with organizations such as the Copyright
Clearance Center and the Copyright Licensing Agency, can be found at our website:
www.elsevier.com/permissions
This book and the individual contributions contained in it are protected under copyright
by the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and
experience broaden our understanding, changes in research methods, professional
practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge
in evaluating and using any information, methods, compounds, or experiments
described herein. In using such information or methods they should be mindful of their
own safety and the safety of others, including parties for whom they have a professional
responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or
editors, assume any liability for any injury and/or damage to persons or property as a
matter of products liability, negligence or otherwise, or from any use or operation of
any methods, products, instructions, or ideas contained in the material herein.
ISBN: 978-0-12-801370-0
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
For information on all Academic Press Publications
visit our website at http://store.elsevier.com/
Printed and bound in the USA
Digital Assets
Thank you for selecting Academic Press’ Bayesian Data Analysis in Ecology
Using Linear Models with R, BUGS, and Stan. To complement the learning
experience, we have provided a number of online tools to accompany this
edition.
To view the R-package “blmeco” that contains all example data and some
specific functions presented in the book, visit www.r-project.org.
The full R-Code and exercises for each chapter are provided at
www.oikostat.ch/blmeco.htm.
x
Acknowledgments
The basis of this book is a course script written for statistics classes at the
International Max Planck Research School
for Organismal Biology
(IMPRS)dsee www.orn.mpg.de/2453/Short_portrait. We, therefore, sincerely
thank all the IMPRS students who have used the script and worked with us.
The text grew as a result of questions and problems that appeared during the
application of linear models to the various Ph.D. projects of the IMPRS stu-
dents. Their enthusiasm in analyzing data and discussion of their problems
motivated us to write this book, with the hope that it will be of help to future
students. We especially thank Daniel Piechowski and Margrit Hieber-Ruiz for
hiring us to give the courses at the IMPRS.
The main part of the book was written by FK and PK during time spent at
the Colorado Cooperative Fish and Wildlife Research Unit and the Department
of Fish, Wildlife, and Conservation Biology at Colorado State University in
the spring of 2014. Here, we were kindly hosted and experienced a motivating
time. William Kendall made this possible, for which we are very grateful.
Gabriele Engeler and Joyce Pratt managed the administrational challenges of
tenure there and made us feel at home. Allison Level kindly introduced us to
the CSU library system, which we used extensively while writing this book.
We enjoyed a very inspiring environment and cordially thank all the Fish,
Wildlife, and Conservation Biology staff and students who we met during
our stay.
The companies and institutions at which the authors were employed during
the work on the book always positively supported the project, even when it
produced delays in other projects. We are grateful to all our colleagues at
the Swiss Ornithological
Institute (www.vogelwarte.ch), oikostat GmbH
(www.oikostat.ch), Hintermann & Weber AG (www.hintermannweber.ch), the
University of Basel, and the Clinical Trial Unit at the University of Basel
(www.scto.ch/en/CTU-Network/CTU-Basel.html).
We are very grateful to the R Development Core Team (http://www.
r-project.org/contributors.html) for providing and maintaining this wonderful
software and network tool. We appreciate the flexibility and understandability
of the language R and the possibilitiy to easily exchange code. Similarly, we
would like to thank the developers of BUGS (http://www.openbugs.net/w/
BugsDev) and Stan (http://mc-stan.org/team.html)
their
extremely useful software freely available. Coding BUGS or Stan has helped
for making all
xi
xii Acknowledgments
us in many cases to think more clearly about the biological processes that
have generated our data.
Example data were kindly provided by the Ulmet-Kommission (www.bnv.
ch), the Landschaft und Gewa¨sser of Kanton Aargau, the Swiss Ornithological
Institute (www.vogelwarte.ch), Valentin Amrhein, Anja Bock, Christoph
Bu¨hler, Karl-Heinz Clever, Thomas Gottschalk, Martin Gru¨ebler, Gu¨nther
Herbert, Thomas Hoffmeister, Rainer Holler, Beat Naef-Daenzer, Werner
Peter, Luc Schifferli, Udo Seum, Maris Strazds, and Jean-Luc Zollinger.
For comments on the manuscript we thank Martin Bulla, Kim Meichtry-
Stier and Marco Perrig. We also thank Roey Angel, Karin Boos, Paul Conn,
and two anonymous reviewers for many valuable suggestions regarding the
book’s structure and details in the text. Holger Schielzeth gave valuable
comments and input for Chapter 10, and David Anderson and Michael Schaub
commented on Chapter 11. Bob Carpenter figured out essential parts of the
Stan code for the CormackeJollyeSeber model. Michael Betancourt and Bob
Carpenter commented on the introduction to MCMC and the Stan examples.
Valentin Amrhein and Barbara Helm provided input for Chapter 17. All
these people greatly improved the quality of the book, made the text more
accessible, and helped reduce the error rate.
Finally, we are extremely thankful for the tremendous work that Kate
Huyvaert did proofreading our English.
Chapter 1
Why do we Need Statistical
Models and What is this Book
About?
Chapter Outline
1.1 Why We Need Statistical
Models
1
1.2 What This Book is About
2
1.1 WHY WE NEED STATISTICAL MODELS
There are at least four main reasons why statistical models are used: (1) models
help to describe how we think a system works, (2) data can be summarized using
models, (3) comparison of model predictions with data helps with understanding
the system, and (4) models allow for predictions, including the quantification
of their uncertainty, and, therefore, they help with making decisions.
A statistical model is a mathematical construct based on probability theory
that aims to reconstruct the system or the process under study; the data are
observations of this system or process. When we speak of “models” in this book,
we always mean statistical models. Models express what we know (or, better,
what we think we know) about a natural system. The difference between the
model and the observations shows that what we think about the system may still
not be realistic and, therefore, points out what we may want to think about more
intensively. In this way, statistical models help with understanding natural
systems.
Analyzing data using statistical models is rarely just applying one model to
the data and extracting the results. Rather, it is an iterative process of fitting a
model, comparing the model with the data, gaining insight into the system from
specific discrepancies between the model and the data, and then finding a more
realistic model. Analyzing data using statistical models is a learning process.
Reality is usually too complex to be perfectly represented by a model. Thus, no
model is perfect, but a good model is useful (e.g., Box, 1979). Often, several
models may be plausible and fit the data reasonably well. In such cases, the
inference can be based on the set of all models, or a model that performs best for
Bayesian Data Analysis in Ecology Using Linear Models with R, BUGS, and Stan
http://dx.doi.org/10.1016/B978-0-12-801370-0.00001-0. Copyright © 2015 Elsevier Inc. All rights reserved.
1
2 Bayesian Data Analysis in Ecology Using Linear Models with R, BUGS, and Stan
a specific purpose is selected. In Chapter 11 we have compiled a number of
approaches we found useful for model comparisons and multimodel inference.
Once we have one or several models, we want to draw inferences from the
model(s). Estimates of the effects of the predictor variables on the outcome
variables, fitted values, or derived quantities that are of biological interest are
extracted, together with an uncertainty estimate. In this book we use, except in
one example, Bayesian methods to assess uncertainty of the estimates.
Models summarize data. When we have measured the height of 100 trees in
a forest and we would like to report these heights to colleagues, we report the
mean and the standard deviation instead of reporting all 100 values. The mean
and the standard deviation, together with a distributional assumption (e.g., the
normal distribution) represent a statistical model that describes the data. We do
not need to report all 100 values because the 2 values (mean and standard
deviation) describe the distribution of the 100 values sufficiently well so that
people have a picture of the heights of the 100 trees. With increasing
complexity of the data, we need more complex models that summarize the
data in a sensible way.
Statistical models are widely applied because they allow for quantifying
uncertainty and making predictions. A well-known application of statistical
models is the weather forecast. Additional examples include the prediction of
bird or bat collision risks at wind energy turbines based on some covariates,
the avalanche bulletins, or all the models used to predict changes of an
ecosystem when temperatures rise. Political decisions are often based on
models or model predictions. Models are pervasive; they even govern our daily
life. For example, we first expected our children to get home before 3:30 p.m.
because we knew that the school bus drops them off at 3:24, and a child can
walk 200 m in around 4 min. What we had in mind was a model child. After
some weeks observing the time our children came home after school, we could
compare the model prediction with real data. Based on this comparison and
short interviews with the children, we included “playing with the neighbor’s
dog” in our model and updated the expected arrival time to 3:45 p.m.
1.2 WHAT THIS BOOK IS ABOUT
This book is about a broad class of statistical models called linear models.
Such models have a systematic part and a stochastic part. The systematic part
describes how the outcome variable (y, variable of interest) is related to the
predictor variables (x, explanatory variables). This part produces the fitted
values that are completely defined by the values of the predictor variables. The
stochastic part of the model describes the scatter of the observations around
the fitted values using a probability distribution. For example, a regression
line is the systematic part of the model, and the scatter of the data around
the regression line (more precisely: the distribution of the residuals) is the
stochastic part.
Why do we Need Statistical Models Chapter j 1 3
Linear models are probably the most commonly used models in biology and
in many other research areas. Linear models form the basis for many statistical
methods such as survival analysis, structural equation analysis, variance
components analysis, time-series analysis, and most multivariate techniques. It
is of crucial importance to understand linear models when doing quantitative
research in biology, agronomy, social sciences, and so on. This book introduces
linear models and describes how to fit linear models in R, BUGS, and Stan. The
book is written for scientists (particularly organismal biologists and ecologists;
many of our examples come from ecology). The number of mathematical
formulae is reduced to what we think is essential to correctly interpret model
structure and results.
Chapter 2 provides some basic information regarding software used in this
book, important statistical terms, and how to work with them using the statistical
software package R, which is used in most chapters of the book.
The linear relationship between the outcome y and the predictor x can be
straightforward, as in linear models with normal error distribution (normal linear
model, LM, Chapter 4). But the linear relationship can also be indirect via a link
function. In this case, the direct linear relationship is between a transformed
outcome variable and the predictor variables, and, usually, the model has a
nonnormal error distribution such as Poisson or binomial (generalized linear
model, GLM, Chapter 8). Generalized linear models can handle outcome
variables that are not on a continuous and infinite scale, such as counts and
proportions.
For some linear models (LM, GLM) the observations are required to be
independent of each other. However, this is often not the case, for example, when
more than one measurement is taken on the same individual (i.e., repeated
measurements) or when several individuals belong to the same nest, farm, or
another grouping factor. Such data should be analyzed using mixed models
(LMM, GLMM, Chapters 7 and 9); they account for the nonindependence of
the observations. Nonindependence of data may also be introduced when
observations are made close to each other (in space or time). In Chapter 6 we
show how temporal or spatial autocorrelation is detected and we give a few hints
about how temporal correlation can be addressed. In Chapter 13, we analyze
spatial data using a species distribution example.
Chapter 14 contains examples of more complex analyses of ecological data
sets. These models should be understandable with the theory learned in the first
part of the book. The chapter presents ideas on how the linear model can be
expanded to more complex models. The software BUGS and Stan, introduced in
Chapter 12, are used to fit these complex models. BUGS and Stan are relatively
easy to use and flexible enough to build many specific models. We hope that this
chapter motivates biologists and others to build their own models for the
particular process they are investigating.
Throughout the book, we treat model checking using graphical methods
with high importance. Residual analysis is discussed in Chapter 6. Chapter 10
相关推荐
2023年江西萍乡中考道德与法治真题及答案.doc
2012年重庆南川中考生物真题及答案.doc
2013年江西师范大学地理学综合及文艺理论基础考研真题.doc
2020年四川甘孜小升初语文真题及答案I卷.doc
2020年注册岩土工程师专业基础考试真题及答案.doc
2023-2024学年福建省厦门市九年级上学期数学月考试题及答案.doc
2021-2022学年辽宁省沈阳市大东区九年级上学期语文期末试题及答案.doc
2022-2023学年北京东城区初三第一学期物理期末试卷及答案.doc
2018上半年江西教师资格初中地理学科知识与教学能力真题及答案.doc
2012年河北国家公务员申论考试真题及答案-省级.doc
2020-2021学年江苏省扬州市江都区邵樊片九年级上学期数学第一次质量检测试题及答案.doc
2022下半年黑龙江教师资格证中学综合素质真题及答案.doc
